Research On Numerical Solution Of Differential Equation Based On Neural Network

Differential equations is an important mathematical and physical model in the development of science.It depicts the basic laws of nature in mathematical language.Since many differential equations cannot be solved analytically,the study of numerical solutions of differential equations plays an important role in mathematics and physics.Numerical methods for solving differential equations mainly include finite difference method,finite element method and spectral method.This paper will study the method based on neural network to solve the numerical solution of differential equations,mainly including the adaptive finite element mesh method based on neural network and the deflation method of multi-solution problem of nonlinear differential equations.In the finite element solution method of differential equations,the overall error of the finite element solution will be affected by the local low regularity of the exact solution of the problem.The low regularity of the solution may come from the discontinuity of the coefficients in the equation or the complexity of the definition region and other factors.In order to obtain a numerical solution that meets the accuracy requirements,an intuitive idea is to subdivide the local low-regularity region of the solution,which is the main idea of the grid adaptation method.So far,the grid adaptive methods mainly include adaptive finite element method,adaptive moving grid method and so on.In this paper,the numerical solution of the differential equation is expressed as a function of the grid parameters,and the corresponding functional minimization problem is established.Then the neural network algorithm is utilized to update the grid parameters in order to minimize the energy functional,so as to achieve the purpose of grid adaptation.In practical applications,there are many physical models corresponding to nonlinear differential equations.However,the solutions of many nonlinear differential equations are not unique.How to find all the solutions of nonlinear differential equations is the key problem in applying these models.Common methods for numerically solving multisolution problems include Newton-Raphson method,deflation method and so on.In this paper,the loss function is constructed based on the energy function of the differential equation to obtain the partial solution of the deflation method,and then the modified loss function of the neural network is constructed by the deflation operator based on the partial solution,so as to obtain a neural network for solving the multi-solution problem of nonlinear differential equations deflation numerical methods.